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Abstract

This paper discusses abductive reasoning---that is, reasoning in which
explanatory hypotheses are formed and evaluated. First, it criticizes two
recent formal logical models of abduction. An adequate formalization would
have to take into account the following aspects of abduction: explanation
is not deduction; hypotheses are layered; abduction is sometimes creative;
hypotheses may be revolutionary; completeness is elusive; simplicity is
complex; and abductive reasoning may be visual and non-sentential. Second,
in order to illustrate visual aspects of hypothesis formation, the paper
describes recent work on visual inference in archaeology. Third, in connection
with the evaluation of explanatory hypotheses, the paper describes recent
results on the computation of coherence.

There has been an explosion of recent work in artificial intelligence
that recognizes the importance of abductive reasoning---that is, reasoning
in which explanatory hypotheses are formed and evaluated. Many important
kinds of intellectual tasks, including medical diagnosis, fault diagnosis,
scientific discovery, legal reasoning, and natural language understanding
have been characterized as abduction. Appropriately, attempts have been
made to achieve a more exact understanding of abductive reasoning by developing
formal models that can be used to analyze the computational properties of
abductive reasoning and its relation to other kinds of inference. Bylander
et al.[4] have used their formal model to analyze the computational
complexity of abduction and show that in general it is NP-hard. Konolige
[16] has used a similar formal model to derive results concerning the relation
between abductive reasoning and Reiter's model of diagnosis [16]. While
these formal results are interesting and useful, it would be unfortunate
if researchers were to conclude that the analyses of Konolige, Bylander
et al. have provided a precise understanding of abductive reasoning.
We shall discuss numerous important aspects of inference to explanatory
hypotheses that are not captured by one or both of the formalisms that these
authors have proposed. In particular, we show how these models do not adequately
capture abductive discovery using representations that are pictorial, and
we argue that abductive evaluation should be conceived in terms of coherence
rather than deduction.

1. The models

Konolige's model of abduction is contained in two definitions:

Definition 1.1 Let L be a first order language. A simple causal
theory is a tuple {C,E,Sigma} where:

1. C, a set of sentences of L, are the causes;

2. E, a set of sentences of L, are the effects;

3. Sigma, a set of sentences of L, is the domain theory.

Definition 1.2Let {C,E,Sigma} be a simple causal theory. An
explanation of a set of observations O subset E is a finite set A subset
C such that:

1. A is consistent with Sigma;

2. Sigma cup A vdash O;

3. A is subset-minimal over sets satisfying the first two conditions,
i.e., there is no proper subset of A consistent with Sigma that implies
O.

Bylander and his colleagues offer a similar definition:

Definition 1.3An abduction problem is a tuple {Dall,Hall,e,pl}
where:

1. Dallis a finite set of all the data to be explained;

2. Hallis a finite set of all the individual hypotheses;

3. e is a map from all subsets of Hallto subsets of
Dall;

4. pl is a map from subsets of Hall to a partially ordered
set representing the plausibility of various hypotheses.

A set of hypotheses H is an explanation if it is complete
and parsimonious, i.e., if e(H) = Dall, and no proper subset
of H explains all the data that H does.

The similarities between these two sets of definitions are clear. Konolige's
[16] causes correspond to Bylander et al. [4] hypotheses; Konolige's
effects correspond to Bylander et al.'s data. Konolige's requirement
that a set of causes be subset-minimal is the same as Bylander et al.'s
requirement that a set of hypotheses be parsimonious. (Peng and Reggia [21]
call the same property irredundancy.) Bylander et al. [4] go beyond
Konolige in requiring an explanation to explain all the data and in adding
a plausibility ordering on hypotheses. On the other hand, their definition
is more general than Konolige's in not restricting data and hypotheses to
sentences.

These definitions usefully capture much of what is central in abductive
reasoning, particularly the goal of assembling a set of hypotheses (causes)
that provide good explanations of the data (effects). But each of them oversimplifies
abductive reasoning in several important respects.

2. The limitations

2.1 Explanation is not deduction

First, let us consider the nature of explanation. The analysis of Bylander
et al. contains the unanalyzed function e that specifies which
hypotheses explain which data. Konolige's account has the apparent advantage
that it spells out this relation by using the notion of logical consequence
as represented by the turnstile "vdash". Hypotheses (causes) explain
data (effects) if the latter can be deduced from the former and the domain
theory. Konolige is thus assuming a deductive account of explanation that
has been the subject of several decades of critical discussion in the philosophy
of science.

In 1948, Hempel and Oppenheim proposed what has come to be known as the
deductive-nomological model of explanation [12]. On this model, an
explanation is a deductive argument in which a sentence representing a phenomenon
to be explained is derived from a set of sentences that describe particular
facts and general laws (nomos is Greek for law). This model provides
a good approximation for many explanations, particularly in mathematical
physics. But it is evident that the deductive model fails to provide either
necessary or sufficient conditions for explanation. See [15] for a comprehensive
review of several decades of philosophical discussions of the nature of
explanation, and [18] for application to artificial intelligence.

First, there are many explanations in science and everyday life that
do not conform to the deductive model. Hempel himself discussed at length
statistical explanations in which what is explained follows only probabilistically,
not deductively, from the laws and other sentences that do the explaining.
Many critics have argued that explanations in such fields as history and
evolutionary biology rarely have a deductive form. In both philosophy and
AI, researchers have proposed that many explanations can be understood in
terms of applications of schemas that fit a phenomenon into a pattern without
producing a deductive argument. (For a review of different approaches to
explanation in these fields, see [27, 28].) For example, a Darwinian explanation
of how a species evolved by natural selection applies a general pattern
that cites biological mechanisms and historical facts to suggest how an
adaptation might have come about. But the historical record is too sparse
and biological principles are too qualitative and imprecise for deductive
derivation. Thus Konolige's use of deduction in his characterization of
abduction arbitrarily excludes many domains in which hypotheses are formed
and evaluated but do not provide deductive explanations.

Second, the deductive model of explanation does not even provide sufficient
conditions for explanation, since there are examples that conform to the
model but do not appear to constitute explanations. For example, we can
deduce the height of a flagpole from information about its shadow along
with trigonometry and laws of optics. But it seems odd to say that the length
of a flagpole's shadow explains the flagpole's height. Konolige's subset-minimality
requirement serves to rule out some of the cases of irrelevance that philosophers
have discussed, for example the explanation that a man is not pregnant because
he takes birth control pills. But other examples such as the flagpole show
that some additional notion of causal relevance is crucial to many kinds
of explanation, and there is little hope of capturing this notion using
logic alone. Contrast Pearl's [20] work on Bayesian networks and Peng and
Reggia's [21] model of abduction which employ ineliminably intuitive notions
of causality. A general model of abduction requires an account of explanation
that is richer than deduction.

2.2 Hypotheses are layered

Bylander et al. assume that abduction can be characterized by
distinguishing sets of data from sets of hypotheses, with explanation consisting
of a mapping from the latter to the former. This characterization is adequate
for straightforward kinds of medical diagnosis involving a given set of
symptoms (data) that can be explained by a given set of diseases. But it
neglects the complexity of other kinds of abductive tasks in which the organization
of hypotheses and data is more complex. In particular, it rules out a large
set of cases where hypotheses explain other hypotheses and selection of
the best overall explanation depends on taking these relations into account.
For example, in legal inference the availability of a possible motive goes
a long way to increasing the chance for a murder conviction. The data are
the various pieces of physical evidence, e.g., that the suspects'
fingerprints were found on the murder weapon. The hypothesis is that the
suspect is the murderer, and a higher level hypothesis might be that the
suspect hated the victim because of some previous altercation. The plausibility
of the lower-level hypothesis comes not only from what it explains, but
also from it itself being explained. This kind of hierarchical explanation
in which hypotheses explain other hypotheses that explain data is also found
in science and medicine; it has been discussed both in the context of Bayesian
models of probabilistic belief revision [20] and in the context of explanatory
coherence accounts of scientific inference [26, 28].

Just as Bylander et al. assume that only data are explained, so
Konolige assumes that only observed effects are explained by derivation
from causes. But as both Bayesian and explanatory coherence analyses allow,
causes are often themselves effects and assessment of overall acceptability
of explanatory hypotheses must take this into account.

2.3 Abduction is sometimes creative

The definitions of Konolige and Bylander et al. have the unfortunate
implication that abduction is always a matter of selection from a known
set of hypotheses or causes. Creative abduction often involves the construction
of novel hypotheses involving newly formed concepts such as natural selection
or AIDS. For Charles Peirce, who coined the term ``abduction'' a century
ago, the introduction of unifying conceptions was an important part of abduction
[11, 25], and it would be unfortunate if our understanding of abduction
were limited to more mundane cases where hypotheses are simply assembled.
Abduction does not occur in the context of a fixed language, since the formation
of new hypotheses often goes hand in hand with the development of new theoretical
terms such as ``atom,'' ``electron,'' ``quark,'' ``gene,'' ``neuron'' and
``AIDS.''

2.4 Hypotheses may be revolutionary

The first clause of Konolige's second definition requires that a hypothesized
cause A be consistent with a domain theory. While this requirement may be
acceptable for mundane applications, it will not do for interesting cases
of belief revision where the introduction of new hypotheses leads to rejection
of previously held theories. For example, when Darwin proposed the hypothesis
that species evolve as the result of natural selection, he was contradicting
the dominant view that species were fixed by divine creation. Konolige's
consistency requirement would prevent Darwin's hypothesis from ever being
considered as part of an abductive explanation that would eventually supplant
the accepted biological/theological domain theory. We can not simply delete
a belief and then replace it with one inconsistent with it, because until
the new belief comes in competition with the old one, there is no reason
to delete the old one. Belief revision requires a complex balancing of a
large number of beliefs and kinds of evidence. Algorithms for performing
such balancing are available, such as Bayesian belief revision and connectionist
judgments of explanatory coherence. Bylander et al. do not have a
consistency condition like Konolige's.

2.5 Completeness is elusive

Bylander et al. do not count a set of hypotheses as an explanation
unless it is complete, i.e., explains all the data. They admit that
there might be ``partial'' explanations, but want completeness as a requirement
because of the goal that all the data should be explained. This is a laudable
goal, but does not justify building completeness into the definition of
an explanation, since the goal is so rarely accomplished in realistic situations.
From medicine to science, it is not typically the case that everything can
be explained even by the best of theories. Even the greatest scientific
theories such as Newtonian mechanics have faced anomalies through their
histories; for example, the perihelion of Mercury was not explained until
Einstein developed the theory of relativity. Doctors need to make diagnoses
even when some symptoms remain puzzling. The requirement of completeness
makes sense only in limited closed domains such as simple circuits where
one can be assured that everything can be explained given the known
causes.

2.6 Simplicity is complex

Many researchers on abduction have seen the need to use simplicity as
one of the constraints on inference to the best explanation, i.e.,
we want not simply a set of hypotheses that explain as much as possible,
but also the most economical set of assumptions possible. Hobbs et al.
[13] sum this up by saying that we want the most bang (facts explained)
for the buck (quantity of assumptions). But simplicity is an elusive notion
that is not adequately captured by the relation of subset minimality which
deals only with cases where we can prefer an explanation by a set of hypotheses
to an explanation by a superset of those hypotheses. We need a broader notion
of simplicity to handle cases where the competing explanations are accomplished
by sets of hypotheses that are of different sizes but are not subsets of
each other. Such cases are common in the history of science and require
richer notions of simplicity than subset minimality [28, 25].

2.7 Abductive reasoning may use non-sentential representations

Konolige's definition, like virtually all available accounts of abduction,
presumes that explanatory hypotheses are represented sententially. But we
will now show that some abductive inference is better understood as using
pictorial or other iconic representations.

3. Visual abduction

Peirce claimed that all thinking is in signs, and that signs can be icons,
indices, or symbols [10, 11]. Icons are signs that resemble what they represent;
examples include pictures, photographs, and geometrical diagrams. Peirce
placed great emphasis on diagrammatic thinking and even developed a powerful
system of predicate logic based on diagrams or ``existential graphs'' [23].
Surprisingly, Peirce does not seem to have connected his two highly original
insights: we know of no text in which he discusses abduction as diagrammatic
or iconic. But there are instances of abductive thinking that are most plausibly
interpreted as pictorial.

Suppose you return to your car at the shopping center and find a big
scratch on one door. Naturally, you wonder how it happened and start to
generate hypotheses to explain how the scratch came to be. Your abductions
may be purely verbal, if you start to apply rules such as ``If a car door
is opened and collides with another car door, the latter door is scratched.''
You could then verbally abduce that some other car door was opened and collided
with yours. But a mode of thinking that is natural for many people is to
perform the same kind of thinking pictorially. You can form a mental image
of a car driving up beside yours and then its driver opening a door that
scratches yours. Here the explanation is a kind of mental movie in which
you imagine your door being scratched. The abductive inference that the
accident happened this way involves a mental picture of the other car's
door hitting yours. Such pictures provide an iconic representation of the
event that you conjecture to have happened, since the picture you form resembles
the hypothesized event in a much more direct way than a verbal/sentential
representation would. Whenever our knowledge of how things work in the world
involves dynamic pictorial representations, these representations can be
used to generate iconic explanations of what occurs. Many scientists have
reported that images played a crucial role in their most creative thinking:
the most eminent include Bohr, Boltzmann, Einstein, Faraday, Feynman, Heisenberg,
Helmholtz, Herschel, Kekule, Maxwell, Poincare, Tesla, Watson, and Watt.

Performing abduction visually may have strong cognitive advantages. With
verbal representations such as rules, it may be necessary to search through
many possible inferences before finding a plausible explanatory hypothesis.
But a picture of a situation may immediately suggest a likely cause, if
it vividly displays factors that are spatially contiguous and therefore
more likely to be causally relevant. Artificial intelligence is still very
limited in its ability to use such visuospatial information, although progress
is being made in the direction of imagistic representations. Glasgow has
argued for the importance of spatial reasoning in AI and proposed a representational
scheme based on 3-dimensional arrays [9, 8]. Graphs provide visual representations
that are more flexible than arrays [5, 30], so to show how abduction can
be visual yet nevertheless amenable to formal treatment, we will now discuss
graph grammars.

A simple graph can be thought of as a purely algebraic structure consisting
of a set of vertices and edges, but uses of graphs often exploit their visual
character. When we draw a graph representing, for example, the paths between
various cities, we get a diagram or mental representation of the routes
between cities that resembles the actual roads. The mathematical structure
of a graph naturally translates into a graphical diagram that resembles
what it represents much more directly than a set of sentences would. Graph
grammars consist of sets of production rules that differ from standard verbal
productions in that the left-hand sides (conditions) and right-hand sides
(actions) are represented as graphs rather than as verbal structures [7,
14, 19]. A graphical production can be interpreted as saying that if you
have one graphical structure and you apply a transformation to it, then
you get another graphical structure. For abduction purposes, we can think
of the right-hand side of a graph production as providing a visual representation
of something to be explained, and the left-hand side and the transformation
as providing a possible visual explanation.

(Graphic not available)

Figure 1. Graph representation of a Lego block.

To be more concrete, consider children's interlocking Lego blocks. We
can represent each block as a graph whose vertices are connectors and sockets,
where the connectors on one block fit into the sockets on another block.
Figure 1 gives a graph for a simple 4-connector block, with 8 vertices and
12 edges. Transformations possible for Lego blocks include stacking one
on top of the other, which produces a new structure in which the connectors
on the bottom block go into the sockets on the top block. A graphical production
representing this would have a left-hand side of two unconnected graphs
and a transformation that produced new edges connecting the appropriate
sockets and connectors, producing a new connected graph with a total of
16 vertices and 28 edges, including 4 new ones.

Given such a production, we could explain the existence of a tower consisting
of two blocks by hypothesizing that there were two independent blocks that
had been transformed by joining. Abduction is then visual because the representations
of both what gets explained and what does the explaining use structures
that resemble what they represent.

Let us now state this more formally.

Definition 3.1A graph G is a tuple {V,E} where:

V is a set of vertices;

E subset (V times V) is a set of edges.

Definition 3.2A graph grammar Gamma is a finite set of productions
P, where a production is a tuple {Gl,Gr,T} such
that:

Gl is the left-hand side;

Gr is the right-hand side;

T is the embedding transformation that specifies the relations between
the vertices and edges of Gl and Gr.

Definition 3.3A graph-grammatical abductive explanation of
a target graph G^tis a hypothesis graph G^h such that there is a
set of productions in P whose successive transformations transform G^h into
G^t.

Intuitively, the hypothesis graph G^h provides an explanation of the
target graph G^t by virtue of a series of transformations that show how
the target graph can be produced from the hypothesis graph.

Graph grammars are not the only possible basis for visual abduction.
Leyton [17] presented a ``process grammar'' that could be used to infer
the causal history of objects from their shapes. This grammar contains such
productions as one that can be interpreted: ``If a shape is squashed, it
will indent.'' If the shapes and the processes of squashing and indenting
are represented pictorially, then the inference that explains a shape's
indentation by its having been squashed is an instance of visual abduction.

This paper does not attempt to present a general theory of visual abduction,
but rather some instances that show the limitations of current formal models
of abductive reasoning. A fully general characterization of abduction would
have to allow representations of causes and effects that are pictorial as
well as ones that are sentential. It would also have to admit forms of explanation
that centrally employ visual transformations as well as ones that employ
deduction and other verbal processes. In addition, we should not rule out
the possibility of a multimodal theory of abduction that includes non-visual,
non-verbal representations involving smell, touch, and emotion. For example,
there are some diseases that physicians can diagnose based on a patient
having a particular odor. The need for a multimodal theory is illustrated
below with an archaeological example.

4. The demise of SK54

Forming hypotheses to explain the unusual macroscopic properties of artifacts
and skeletal remains is an important part of archaeology. This kind of abduction
often requires the archaeologist to reconstruct the events in the history
of an object which caused it to change in shape or structure from some initial
form to the final, observed one. Since this kind of evidence for the history
of an object is visual in nature, the archaeologist may find visual mental
imagery useful in generating an explanation of it. (See [24] for further
discussion.

Figure 3. The abduced fate of SK54 in the jaws of a leopard [2,
p. 1118]

For example, in 1949 excavations of a cave at Swartkrans in South Africa
yielded, among much other debris, the skullcap of an australopithecine (thereafter
designated SK54). The distinctive features of this skullcap consisted
of two notches, one on either side of the centerline, which had obviously
been driven into the back of the skull by two pointed objects when the creature
was still alive [2]. SK54 is pictured in figure 2, with the notches indicated
by the two arrows. At first it was supposed that the notches had been inflicted
by two separate blows from a weapon wielded by another hominid, because
each notch had been made at divergent angles from the centerline
[1]. This hypothesis accorded well with the prevailing theory that human
evolution had been driven by murder and cannibalism---Dart's [6] ``killer
ape'' hypothesis.

However, an alternative explanation has been offered by Brain [2]. Noting
that the lower canine teeth of leopards diverge and are about the right
distance apart, Brain hypothesized that the notches had been created by
a leopard which had taken the australopithecine's head in its mouth, as
dramatically illustrated in figure 3. Indeed, a fossil leopard jaw from
the same site (SK349) fits the notches fairly well and shows that the hypothesis
is a plausible one; see also [3]. This is shown in figure 4.

This explanation of the notches in SK54 also accords with several other
facts about the debris in the Swartkrans cave. The entrance to the cave
was a vertical shaft when the australopithecine remains were deposited and
those remains are mostly comprised of skull fragments. Similar shaft caves
in the area today are frequented by leopards which use the trees which grow
around the entrances as places to safely consume prey out of the reach of
hyenas. Since leopards tend to destroy the skeletal material of their primate
prey with the exception of the skulls, the leopard-predator hypothesis would
also explain why the australopithecine remains are mostly skull fragments---the
skulls would simply have fallen from the trees and into the cave shafts
[2]. This scenario, leopard predation of early hominids, is much different
from the ``killer ape'' scenario favored by Dart.

Brain's leopard hypothesis exemplifies the use of visual abduction. The
target of explanation---the unusual notches in the skullcap SK54---are highly
visual in nature, consisting in their placement, depth, and direction. The
hypothesis reconstructs the vital moment in the history of an unfortunate
hominid, when its head was clenched in the jaws of a leopard to produce
the notches. Because the relevant data are spatial, the hypothesis is most
parsimoniously captured visually, as in figure 3, and may well have first
occurred to Brain as just such a mental picture. The hypothesis thus abduced
was then corroborated by further evidence of fossil leopard jaws and the
feeding habits of modern leopards. Thus, this example illustrates that visual
abduction fits the expanded criteria for abduction discussed in section
2.

5. Coherence

Abduction is often thought of as a kind of reverse deduction; see section
2.1. Whereas modus ponens infers from p right arrow q and p to
q, abduction infers from p right arrow q and q to p. This
formulation trivializes the point of abduction, which is to form and evaluate
hypotheses that make sense of puzzling facts. Making sense is a holistic
matter of fitting the puzzling facts into a coherent pattern of representations.
The notion of coherence has usually remained vague, but Thagard and Verbeurgt
[29] have recently defined coherence in terms of maximization of constraint
satisfaction. After briefly stating their definition, we will show how abduction
can be reconceptualized as a coherence problem.

Let E be a finite set of elements ei and C be a set of constraints
on E understood as a set (ei, ej) of pairs of elements of E. C divides into
C+, the positive constraints on E, and C-, the negative constraints on E.
With each constraint is associated a number w, which is the weight (strength)
of the constraint. The problem is to partition E into two sets, A (accepted)
and R (rejected), in a way that maximizes compliance with the following
two coherence conditions:

1. if (ei, ej) is in C+, then ei is in A if and only if ej is in
A;

2. if (ei, ej) is in C-, then ei is in A if and only if ej is in
R.

Let W be the weight of the partition, that is, the sum of the weights
of the satisfied constraints. The coherence problem is then to partition
E into A and R in a way that maximizes W.

To understand abduction as a coherence problem, we need to specify the
elements and the constraints. The elements are representations of causes
and effects; in line with our discussion of visual abduction, we will allow
the representations to include both sentential and nonsentential representations
such as visual ones. The major positive constraint on abduction is that
if one element explains another, then there is a symmetric positive constraint
between them. In accord with the first of the two coherence constraints
above, this will have the effect that if the explaining element and the
explained element will tend to be accepted or rejected together. If, as
often happens, more than one element is required to explain another element,
the weight on the constraint between each explaining element and the explained
element can be less than if there were only one explaining element, in keeping
with the theory of simplicity of Thagard [28].

Two sorts of negative constraint are possible. If two elements contradict
each other, then there is a symmetric negative constraint between them,
but there is also a negative constraint if two elements offer competing
explanations of the same fact [28]. Deciding which explanatory hypotheses
to accept and which to reject is a matter of maximizing compliance with
the two coherence conditions. Computationally, this is a very difficult
problem and no tractable algorithm for solving it is available, although
various approximation algorithms work quite well [29].

Our account of abduction as a coherence problem avoids all the limitations
we discussed in section 2. We assume that the relation between elements
is explanation, not deduction. Elements can explain other elements which
explain other elements, so hypotheses can be layered and give rise to complex
chains of constraints. Abduction is not defined in terms of a fixed set
of elements but allows the possibility of creation of new elements that
produce a new assessment of coherence. New hypotheses need not be consistent
with existing ones, nor need the facts be completely explained. Maximizing
coherence involves explaining as much as possible (positive constraints)
and being as consistent as possible (negative constraints), but perfection
is not to be sought in abductive reasoning. We are not urging inconsistency
as a general epistemological strategy, only noting that it is sometimes
necessary to form hypotheses inconsistent with what is currently accepted
in order to provoke a general belief revision that can restore consistency.

Reducing the weights on constraints when explanation is accomplished
by multiple hypotheses allows a complex assessment of simplicity. Finally,
since elements can be visual or other nonverbal kinds of representation,
we have transcended the limitation of abductive reasoning to sentences.

6. Conclusion

The definitions of abduction proposed by Konolige and by Bylander, Allemang,
Tanner and Josephson are clearly inadequate as general characterizations
of this important type of reasoning. These definitions made possible precise
comparisons with other models and derivation of complexity theoretic results,
but the results must be viewed as relative to the abstract model of abduction
proposed. One might think that all complexity results would carry over to
models of abduction more elaborate than the ones offered: if abduction is
more complicated than Bylander et al. allow, then the computational
complexity should be at least as great, so abduction would still be NP-hard.
But the main results of Bylander et al. concern the difficulty of
determining whether an explanation exists, where their notion of explanation
is, as we saw, unrealistically restricted to cases where all the data are
to be explained. In scientific reasoning, in contrast, the task is to explain
as much of the data as you can and to come up with a set of explanatory
hypotheses that is better than the current set. It is a matter of satisficing
(choosing the most coherence set of hypotheses) rather than optimizing.
Abduction characterized in terms of coherence is still intractable, but
efficient approximation algorithms exist. There is also the possibility
that heuristic search made possible by visual representations can greatly
improve the computability of abductions, although this remains to be shown.

Thanks to important formal results such as the theory of NP-completeness,
formal methods have achieved much well-deserved prestige in computer science
and artificial intelligence. But an oversimplified formalization can distract
from important aspects of the kinds of reasoning that underlie intelligence.
Graph grammars and other techniques for visual representation offer the
prospects of developing more general accounts of the nature of abductive
reasoning, which should be construed not as a deviant form of deduction,
but as a coherence problem.

Acknowledgements

* We are grateful to John Josephson and Don Roberts for comments on an
earlier draft. Thanks to John Ching and Andrew Wong for ideas about graphical
representations. This research was supported by the National Science and
Engineering Research Council of Canada.